On the Banzhaf-like Value for Cooperative Games with Interval Payoffs

Chunqiao Tan,Wenrui Feng,Weibin Han

doi:10.3390/math8030372

Chunqiao Tan, Wenrui Feng + Show 1 more

Open Access

https://doi.org/10.3390/math8030372

Copy DOI

Abstract

By using Moore’s subtraction operator and a total order on the set of closed intervals, we introduce a new variation of the Banzhaf value for cooperative interval games called the interval Banzhaf-like value which may accommodate the shortcomings of the interval Banzhaf value. We first reveal the relation between this introduced value and the interval Banzhaf value. Then, we present two sets of properties that may be used to determine whether an interval value is median-indifferent to the interval Banzhaf-like value. Finally, in order to overcome the disadvantages of the interval Banzhaf-like value, we propose the contracted interval Banzhaf-like value and give an axiomatization of this proposed value.

Highlights

A cooperative game is represented by a pair ( N, v), where N is a nonempty and finite set of players, and v is a characteristic function from 2 N to R which assigns each coalition a real number
We firstly proposed the interval Banzhaf-like value for cooperative interval games to overcome the shortcomings of the interval Banzhaf value introduced in [13]
The interval Banzhaf-like value is well-defined for every possible interval game, whereas the interval Banzhaf value is only definable for size monotonic interval games

Summary

Introduction

A cooperative game is represented by a pair ( N, v), where N is a nonempty and finite set of players, and v is a characteristic function from 2 N to R which assigns each coalition a real number. The model of cooperative interval games was originally introduced by [7] to handle bankruptcy situations where the estate is known with certainty while claims belong to known bounded intervals of real numbers. Since it has been theoretically developed by a group of researchers around Branzei and Tijs (See [4,8] for details). Shapley value and the interval Banzhaf value were separately proposed by [12,13] for a class of cooperative interval games so called the size monotonic interval games These two values were separately characterized by using a set of properties.

Preliminaries

The Contracted Interval Banzhaf-like Value and Its Axiomatization

Concluding Remarks